A hybrid FD-FV method for first-order hyperbolic conservation laws on Cartesian grids: The smooth problem case

TL;DR

This paper introduces a hybrid finite difference-finite volume method for solving hyperbolic conservation laws on Cartesian grids, emphasizing accuracy and stability in smooth solution regimes without Riemann solvers.

Contribution

The paper develops a novel FD-FV method that combines cell-averaged and nodal values, achieving higher spatial order and linear stability for smooth solutions of hyperbolic PDEs.

Findings

Spatial order is typically one higher than the differential operator.

Methods are linearly stable under CFL conditions.

Numerical tests confirm accuracy and stability across various equations.

Abstract

We present a class of hybrid FD-FV (finite difference and finite volume) methods for solving general hyperbolic conservation laws written in first-order form. The presentation focuses on one- and two-dimensional Cartesian grids; however, the generalization to higher dimensions is straightforward. These methods use both cell-averaged values and nodal values as dependent variables to discretize the governing partial differential equation (PDE) in space, and they are combined with method of lines for integration in time. This framework is absent of any Riemann solvers while it achieves numerical conservation naturally. This paper focuses on the accuracy and linear stability of the proposed FD-FV methods, thus we suppose in addition that the solutions are sufficiently smooth. In particular, we prove that the spatial-order of the FD-FV method is typically one-order higher than that of the discrete differential operator, which is involved in the construction of the method. In addition, the methods are linearly stable subjected to a Courant-Friedrich-Lewy condition when appropriate time-integrators are used. The numerical performance of the methods is assessed by a number of benchmark problems in one and two dimensions. These examples include the linear advection equation, nonlinear Euler equations, the solid dynamics problem for linear elastic orthotropic materials, and the Buckley-Leverett equation.